Homework 3
Problem 1
a) Assume that the temperature is circularly symmetric: u = u(r, t), where r 2 = x2 + y 2.
∂
= kr ∂r
r ∂u
Derive the heat equation in a circular annulus a ≤ r ≤ b without the source, ∂u
∂t
∂r
by writing the fluxes and rate of energy change.
b) Determine the equilibrium temperature if the temperature of the outer circle is T1 and
the inner circle is T2 .
c) Repeat part a) and part b) for two concentric spheres with radii a and b.
Problem 2
For conduction of thermal energy, the heat flux is φ = −K0 ∇u. If in addition the
molecules move at an average velocity v, a process is called convection, then briefly explain
why φ = −K0 ∇u + cρuv. Derive the corresponding equation for heat flow, including both
conduction and convection of thermal energy (assume no source).
Problem 3
Using MATLAB solve the heat conduction problem
∂u
= ∇ · (k(x)∇u)
∂t
in an inhomogeneous bar. Assume the bar has fixed temperature T = 1 at the ends, and
initial temperature T = 0.
a) Assume k = 0.001 if x < 0.6 and k = 1 elsewhere.
b) Assume k = 1 if 0.3 < x < 0.7 and k = 0.001 elsewhere.
c) Assume k = 0.001 everywhere.
Solve the problem till tf inal = 2.
Answer the following questions.
What is the difference between a) and b)? Why?
What is the difference between b) and c)? Why?
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